You can use the square-root algorithm to compute square roots without a calculator. Find the square-root of a number N by solving the equation w2 - N = 0. Approximate solution w from estimate z with

w ~ z - F(z)/S(z) = z - (z2 - N)/(2z) = (z2 + N)/2z

Pick a perfect square near N, and use its square-root as the first estimate, z.

Example: The square-root of 130 can be approximated from estimate z = 11:

w ~ (z2 + N)/2z = [(11)2 + 130]/(2*
11) = 251/22 = 11.40909

The first approximation differs from the true square-root (11.40175) by 0.064%, which is close enough if you have no calculator. If you have a cheap calculator available, a second approximation based on the first approximation will give a better estimate:

w ~ [(11.40909)2 + 130]/(2*11.40909) = 11.40176

The second approximation differs from the true square-root by 0.00002%.

You can find the kth-root of N by solving the equation wk - N = 0. The approximation function is

w ~ z - (zk - N)/(kzk-1)

This approximation doesn't simplify as nicely as the square-root approximation. Approximations will converge faster if the first-estimate is larger than the solution. Therefore, pick an integer for first-estimate z, such that zk > N.